What you’ll learn
Recognize equivalent expressions.
Key idea
An identity establishes equality for every input; one unequal input refutes equivalence.
A finite table illustrates an identity but cannot by itself justify all-input equivalence; one counterexample disproves it.
Worked example
Classify examples and non-examples for the concept 'Recognize equivalent expressions'.
Are 3(x+9) and 3x+9 equivalent for every x? Classify with a distributive-property reason.
- Distributing gives 3(x+9)=3x+27 for every x.
- At x=0 and x=1 the values are 27 and 30; the identity, not just two trials, proves equivalence.
Answer: Non-example — the distributed constant must be 27, so the constants differ.
Quick Check
Classify examples and non-examples for the concept 'Recognize equivalent expressions'.
Are 7(x+1) and 7x+7 equivalent for every x? Classify with a distributive-property reason.